Finiteness of Higher K -Groups of Orders and Group Rings

In this paper, we prove that if R is the ring of integers in a number field F, A any R-order in a semisimple F-algebra, then K_2n(A), G_2n(A) are finite groups for all positive integers n. Hence, even dimensional higher K- and G-groups of integral grouprings of finite groups are finite. We also show that in odd dimensions, SK_n of integral and p-adic integral grouprings of finite p-groups are also finite p-groups (Received: August 2005)     

Homological properties of Fourier algebras on homogeneous spaces

Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A(G/K) are characterised. Biprojectivity and biflatness for the Fourier algebra A(G/K) are studied. A(G/K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup.     

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then K^G₀(A) @ K^G₀(F){K^G_0(A) \cong K^G_0(F)}, generalizing the analogous result for a dvr by Serre (Inst Hautes études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.     

Picard groups of multiplicative invariants

Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = S ^G . As an application, we produce new polycyclic group algebras with nontrivial torsion in K ₀ . Received: April 25, 1996     

A note on Ki-perfect graphs

Ki-perfect graphs are a special instance of F - G perfect graphs, where F and G are fixed graphs with F a partial subgraph of G. Given S, a collection of G-subgraphs of graph K, an F - G cover of S is a set of T of F-subgraphs of K such that each subgraph in S contains as a subgraph a member of T. An F - G packing of S is a subcollection S′? S such that no two subgraphs in S′ have an F-subgraph in common. K is F - G perfect if for all such S, the minimum cardinality of an F - G cover of S equals the maximum cardinality of an F - G packing of S. Thus K_i-perfect graphs are precisely K_i-1 - K_i perfect graphs. We develop a hypergraph characterization of F - G perfect graphs that leads to an alternate proof of previous results on K_i-perfect graphs as well as to a characterization of F - G perfect graphs for other instances of F and G.     

Multilinear enumerative techniques

When S is a finite set and G a finite group acting on S, we consider the problem of rejecting isomorphs in a G-stable subset of S. In previous work we developed a linear algebraic context for this problem by constructing the finite dimensional vector spaceF ^s whereF is a field of characteristic zero. When S is a finite function space, or a finite direct product of finite function spacesF ^s acquires a multilinear structure By various specializations ofG and S and by applications of results which have appeared elsewhere, identities of Sheehan, deBruijn and P61ya are obtained. Furthermore, these same techniques are applied to examples which do not have a clear resolution using the more common formulas     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

On chief factors of finite groups

Let H and K be normal subgroups of a finite group G and let K≤H. If A is a subgroup of G such that AH=AK or A∩H=A∩K, we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

On some infinite dimensional linear groups

Let F be a field, A be a vector space over F, and GL(F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the present article we begin the study of subgroups G of GL(F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = charF.      

Σ-Uniform structures and Σ-functions. II

We construct a family of Σ-uniform Abelian groups and a family of Σ-uniform rings. Conditions are specified that are necessary and sufficient for a universal Σ-function to exist in a hereditarily finite admissible set over structures in these families. It is proved that there is a set S of primes such that no universal Σ-function exists in hereditarily finite admissible sets \mathbbH\mathbbF(G) \mathbb{H}\mathbb{F}(G) and \mathbbH\mathbbF(K) \mathbb{H}\mathbb{F}(K) , where G = ⊕{Z _p | p ∈ S} is a group, Z _p is a cyclic group of order p, K = ⊕{F _p | p ∈ S} is a ring, and F _p is a prime field of characteristic p.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.     

Finite groups with weakly <Emphasis Type="Italic">S</Emphasis>-quasinormal subgroups

We introduce a new subgroup embedding property in a finite group called weakly S-quasinormality. We say a subgroup H of a finite group G is weakly S-quasinormal in G if there exists a normal subgroup K such that HK ⊴ G and H ∩ K is S-quasinormally embedded in G. We use the new concept to investigate the properties of some finite groups. Some previously known results are generalized.     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.     

The K-admissibility of SL(2, 5)

Let K be a field and let G be a finite group. G is K-admissible if there exists a Galois extension L of K with G=Gal(L/K) such that L is a maximal subfield of a central K-division algebra. We characterize those number fields K such that H is K-admissible where H is any subgroup of SL(2, 5) which contains a S ₂-group. The method also yields refinements and alternate proofs of some known results including the fact that A ₅ is K-admissible for every number field K.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthdayThe first author was partly supported by NSF fellowship DMS-8601130; the second author was partly supported by NSF grant DMS-8806371.     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

Groups with bounded representation degree

Let Gbe a group and let K be a field of characteristic p> 0. If all irreducible representations of the group algebra K[G] have finite degree < n, then we show that G has a subgroup A with |G:A| bounded by a function of nand such that all the irreducible representations of K[A] have degree 1.     

Pseudo-coefficients des séries <Emphasis Type="Italic">κ</Emphasis>-discrètes de GL(<Emphasis Type="Italic">n,F</Emphasis>)

Let F be a non-Archimedean locally compact field, n be an integer ≥ 2, κ be a character of F ^× of finite order dividing n, and G be the group GL(n, F). We prove that if π is an irreducible κ-discrete series of G, then there exist some pseudo-coefficients for the twisted character trace(π ∘ A) where A denotes a non-zero intertwining operator between π ⊗ (κ ∘ det) and π.